Theorem prnz 4780

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4765	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4336	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2106   ≠ wne 2940  Vcvv 3474  ∅c0 4321  {cpr 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-dif 3950  df-un 3952  df-nul 4322  df-sn 4628  df-pr 4630

This theorem is referenced by:  opnz  5472  propssopi  5507  fiint  9320  wilthlem2  26562  upgrbi  28342  wlkvtxiedg  28871  shincli  30602  chincli  30700  spr0nelg  46130  sprvalpwn0  46137